Rank-3 scale theorems: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:guest|guest]] and made on <tt>2013-02-14 01:55:38 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-02-14 11:41:06 UTC</tt>.<br>
: The original revision id was <tt>406883482</tt>.<br>
: The original revision id was <tt>406996878</tt>.<br>
: The revision comment was: <tt></tt><br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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* Every triple [[Fokker blocks|Fokker block]] is max variety 3.
* Every triple [[Fokker blocks|Fokker block]] is max variety 3.
* Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)
* Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)
* Triple Fokker blocks form a trihexagonal tiling on the lattice.
* Triple Fokker blocks form a [[http://en.wikipedia.org/wiki/Trihexagonal_tiling|trihexagonal tiling]] on the lattice.
* A scale imprint is that of a Fokker block if and only if it is the [[product word]] of two DE scale imprints with the same number of notes. See &lt;span style="background-color: #ffffff; color: #1155cc; font-family: arial,sans-serif;"&gt;[[@http://www.springerlink.com/content/c23748337406x463/]]&lt;/span&gt;
* A scale imprint is that of a Fokker block if and only if it is the [[product word]] of two DE scale imprints with the same number of notes. See &lt;span style="background-color: #ffffff; color: #1155cc; font-family: arial,sans-serif;"&gt;[[@http://www.springerlink.com/content/c23748337406x463/]]&lt;/span&gt;
* If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L &gt; m &gt; n &gt; s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s
* If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L &gt; m &gt; n &gt; s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s
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<h4>Original HTML content:</h4>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Rank-3 scale theorems&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Theorems"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Theorems&lt;/h1&gt;
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Rank-3 scale theorems&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Theorems"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Theorems&lt;/h1&gt;
  &lt;ul&gt;&lt;li&gt;Every triple &lt;a class="wiki_link" href="/Fokker%20blocks"&gt;Fokker block&lt;/a&gt; is max variety 3.&lt;/li&gt;&lt;li&gt;Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)&lt;/li&gt;&lt;li&gt;Triple Fokker blocks form a trihexagonal tiling on the lattice.&lt;/li&gt;&lt;li&gt;A scale imprint is that of a Fokker block if and only if it is the &lt;a class="wiki_link" href="/product%20word"&gt;product word&lt;/a&gt; of two DE scale imprints with the same number of notes. See &lt;span style="background-color: #ffffff; color: #1155cc; font-family: arial,sans-serif;"&gt;&lt;a class="wiki_link_ext" href="http://www.springerlink.com/content/c23748337406x463/" rel="nofollow" target="_blank"&gt;http://www.springerlink.com/content/c23748337406x463/&lt;/a&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L &amp;gt; m &amp;gt; n &amp;gt; s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s&lt;/li&gt;&lt;li&gt;Any convex object on the lattice can be converted into a hexagon.&lt;/li&gt;&lt;li&gt;Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
  &lt;ul&gt;&lt;li&gt;Every triple &lt;a class="wiki_link" href="/Fokker%20blocks"&gt;Fokker block&lt;/a&gt; is max variety 3.&lt;/li&gt;&lt;li&gt;Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)&lt;/li&gt;&lt;li&gt;Triple Fokker blocks form a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Trihexagonal_tiling" rel="nofollow"&gt;trihexagonal tiling&lt;/a&gt; on the lattice.&lt;/li&gt;&lt;li&gt;A scale imprint is that of a Fokker block if and only if it is the &lt;a class="wiki_link" href="/product%20word"&gt;product word&lt;/a&gt; of two DE scale imprints with the same number of notes. See &lt;span style="background-color: #ffffff; color: #1155cc; font-family: arial,sans-serif;"&gt;&lt;a class="wiki_link_ext" href="http://www.springerlink.com/content/c23748337406x463/" rel="nofollow" target="_blank"&gt;http://www.springerlink.com/content/c23748337406x463/&lt;/a&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L &amp;gt; m &amp;gt; n &amp;gt; s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s&lt;/li&gt;&lt;li&gt;Any convex object on the lattice can be converted into a hexagon.&lt;/li&gt;&lt;li&gt;Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Unproven Conjectures"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Unproven Conjectures&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Unproven Conjectures"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Unproven Conjectures&lt;/h1&gt;
  &lt;ul&gt;&lt;li&gt;Every rank-3 Fokker block has mean-variety &amp;lt; 4, meaning that some interval class will come in less than 4 sizes.&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
  &lt;ul&gt;&lt;li&gt;Every rank-3 Fokker block has mean-variety &amp;lt; 4, meaning that some interval class will come in less than 4 sizes.&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 11:41, 14 February 2013

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author genewardsmith and made on 2013-02-14 11:41:06 UTC.
The original revision id was 406996878.
The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

=Theorems= 
* Every triple [[Fokker blocks|Fokker block]] is max variety 3.
* Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)
* Triple Fokker blocks form a [[http://en.wikipedia.org/wiki/Trihexagonal_tiling|trihexagonal tiling]] on the lattice.
* A scale imprint is that of a Fokker block if and only if it is the [[product word]] of two DE scale imprints with the same number of notes. See <span style="background-color: #ffffff; color: #1155cc; font-family: arial,sans-serif;">[[@http://www.springerlink.com/content/c23748337406x463/]]</span>
* If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L > m > n > s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s
* Any convex object on the lattice can be converted into a hexagon.
* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.

=Unproven Conjectures= 
* Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes.

Original HTML content:

<html><head><title>Rank-3 scale theorems</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Theorems"></a><!-- ws:end:WikiTextHeadingRule:0 -->Theorems</h1>
 <ul><li>Every triple <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> is max variety 3.</li><li>Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)</li><li>Triple Fokker blocks form a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Trihexagonal_tiling" rel="nofollow">trihexagonal tiling</a> on the lattice.</li><li>A scale imprint is that of a Fokker block if and only if it is the <a class="wiki_link" href="/product%20word">product word</a> of two DE scale imprints with the same number of notes. See <span style="background-color: #ffffff; color: #1155cc; font-family: arial,sans-serif;"><a class="wiki_link_ext" href="http://www.springerlink.com/content/c23748337406x463/" rel="nofollow" target="_blank">http://www.springerlink.com/content/c23748337406x463/</a></span></li><li>If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L &gt; m &gt; n &gt; s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s</li><li>Any convex object on the lattice can be converted into a hexagon.</li><li>Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.</li></ul><br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Unproven Conjectures"></a><!-- ws:end:WikiTextHeadingRule:2 -->Unproven Conjectures</h1>
 <ul><li>Every rank-3 Fokker block has mean-variety &lt; 4, meaning that some interval class will come in less than 4 sizes.</li></ul></body></html>
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